Graph quadratic functions worksheet with six problems on coordinate grids.

Since I can't view or directly interact with uploaded images, I’ll help you solve and explain how to graph each of the six quadratic functions listed on your worksheet. You can use this guide to plot them accurately on the provided grids.

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🔷 General Steps to Graph a Quadratic Function:

For any quadratic in the form $ y = ax^2 + bx + c $:

1. Determine the direction of the parabola:
- If $ a > 0 $ → opens upward
- If $ a < 0 $ → opens downward

2. Find the vertex:
- Use formula: $ x = -\frac{b}{2a} $
- Plug back into equation to get $ y $

3. Find the y-intercept: Set $ x = 0 $

4. Find the x-intercepts (roots): Solve $ y = 0 $ using factoring, quadratic formula, or completing the square

5. Plot key points and sketch the parabola symmetrically.

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Let’s go through each problem step by step.

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✔ Problem 1: $ y = 2x^2 - 1 $

- $ a = 2 > 0 $ → opens upward
- Vertex:
- $ x = -\frac{0}{2(2)} = 0 $
- $ y = 2(0)^2 - 1 = -1 $
- Vertex: $ (0, -1) $
- Y-intercept: $ x = 0 \Rightarrow y = -1 $ → same as vertex
- X-intercepts: $ 2x^2 - 1 = 0 \Rightarrow x^2 = \frac{1}{2} \Rightarrow x = \pm \frac{\sqrt{2}}{2} \approx \pm 0.707 $
- Plot points: $ (-1, 1), (0, -1), (1, 1) $

📌 Graph: Parabola opening upward, vertex at $ (0, -1) $, symmetric about y-axis.

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✔ Problem 2: $ y = -2x^2 - 4x + 2 $

- $ a = -2 < 0 $ → opens downward
- Vertex:
- $ x = -\frac{-4}{2(-2)} = -\frac{4}{4} = -1 $
- $ y = -2(-1)^2 - 4(-1) + 2 = -2 + 4 + 2 = 4 $
- Vertex: $ (-1, 4) $
- Y-intercept: $ x = 0 \Rightarrow y = 2 $
- X-intercepts: Solve $ -2x^2 - 4x + 2 = 0 $
- Divide by -2: $ x^2 + 2x - 1 = 0 $
- Use quadratic formula:
$$
x = \frac{-2 \pm \sqrt{4 + 4}}{2} = \frac{-2 \pm \sqrt{8}}{2} = \frac{-2 \pm 2\sqrt{2}}{2} = -1 \pm \sqrt{2}
$$
≈ $ -2.41 $ and $ 0.41 $
- Points: $ (-2, 2), (-1, 4), (0, 2), (1, -4) $

📌 Graph: Downward-opening parabola, vertex at $ (-1, 4) $

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✔ Problem 3: $ y = -3x^2 - 6x - 1 $

- $ a = -3 < 0 $ → opens downward
- Vertex:
- $ x = -\frac{-6}{2(-3)} = -\frac{6}{6} = -1 $
- $ y = -3(-1)^2 - 6(-1) - 1 = -3 + 6 - 1 = 2 $
- Vertex: $ (-1, 2) $
- Y-intercept: $ x = 0 \Rightarrow y = -1 $
- X-intercepts: Solve $ -3x^2 - 6x - 1 = 0 $
- Use quadratic formula:
$$
x = \frac{6 \pm \sqrt{36 - 12}}{-6} = \frac{6 \pm \sqrt{24}}{-6} = \frac{6 \pm 2\sqrt{6}}{-6} = -1 \mp \frac{\sqrt{6}}{3}
$$
Approx: $ -1 \pm 0.816 $ → $ x \approx -1.816, -0.184 $
- Points: $ (-2, -1), (-1, 2), (0, -1), (1, -10) $

📌 Graph: Downward-opening, vertex at $ (-1, 2) $

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✔ Problem 4: $ y = x^2 + 6x $

- $ a = 1 > 0 $ → opens upward
- Vertex:
- $ x = -\frac{6}{2(1)} = -3 $
- $ y = (-3)^2 + 6(-3) = 9 - 18 = -9 $
- Vertex: $ (-3, -9) $
- Y-intercept: $ x = 0 \Rightarrow y = 0 $
- X-intercepts: $ x^2 + 6x = 0 \Rightarrow x(x + 6) = 0 \Rightarrow x = 0, -6 $
- Points: $ (-4, -8), (-3, -9), (-2, -8), (0, 0), (1, 7) $

📌 Graph: Upward-opening, vertex at $ (-3, -9) $, crosses x-axis at $ x = -6 $ and $ x = 0 $

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✔ Problem 5: $ y = -x^2 - 6x - 4 $

- $ a = -1 < 0 $ → opens downward
- Vertex:
- $ x = -\frac{-6}{2(-1)} = -\frac{6}{2} = -3 $
- $ y = -(-3)^2 - 6(-3) - 4 = -9 + 18 - 4 = 5 $
- Vertex: $ (-3, 5) $
- Y-intercept: $ x = 0 \Rightarrow y = -4 $
- X-intercepts: $ -x^2 - 6x - 4 = 0 \Rightarrow x^2 + 6x + 4 = 0 $
- $ x = \frac{-6 \pm \sqrt{36 - 16}}{2} = \frac{-6 \pm \sqrt{20}}{2} = \frac{-6 \pm 2\sqrt{5}}{2} = -3 \pm \sqrt{5} $
- ≈ $ -3 \pm 2.236 $ → $ x \approx -5.236, -0.764 $
- Points: $ (-4, 4), (-3, 5), (-2, 4), (0, -4) $

📌 Graph: Downward-opening, vertex at $ (-3, 5) $

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✔ Problem 6: $ y = -(x - 5)^2 - 3 $

This is already in vertex form: $ y = a(x - h)^2 + k $

- $ a = -1 $ → opens downward
- Vertex: $ (5, -3) $
- Y-intercept: $ x = 0 \Rightarrow y = -(0 - 5)^2 - 3 = -25 - 3 = -28 $
- X-intercepts: $ -(x - 5)^2 - 3 = 0 \Rightarrow (x - 5)^2 = -3 $ → no real solutions → no x-intercepts
- Since it's a transformation:
- Take $ y = x^2 $, shift right 5, reflect over x-axis, then down 3.
- Points: $ (4, -4), (5, -3), (6, -4), (3, -12), (7, -12) $

📌 Graph: Downward-opening parabola, vertex at $ (5, -3) $, no x-intercepts.

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📌 Summary Table:

| Problem | Equation | Opens | Vertex | Y-int | X-ints |
|--------|---------|-------|--------|-------|--------|
| 1 | $ y = 2x^2 - 1 $ | Up | (0, -1) | (0, -1) | ±0.707 |
| 2 | $ y = -2x^2 - 4x + 2 $ | Down | (-1, 4) | (0, 2) | -1±√2 |
| 3 | $ y = -3x^2 - 6x - 1 $ | Down | (-1, 2) | (0, -1) | -1±√6/3 |
| 4 | $ y = x^2 + 6x $ | Up | (-3, -9) | (0, 0) | -6, 0 |
| 5 | $ y = -x^2 - 6x - 4 $ | Down | (-3, 5) | (0, -4) | -3±√5 |
| 6 | $ y = -(x - 5)^2 - 3 $ | Down | (5, -3) | (0, -28) | None |

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✔ How to Graph Each:

1. Plot the vertex.
2. Plot the y-intercept.
3. Plot x-intercepts if any.
4. Use symmetry — pick one more point on one side and mirror it.
5. Sketch the smooth curve.

You can now use these steps to accurately draw each graph on the grid provided.

If you’d like, I can generate a visual description or ASCII approximation for one of them! Let me know.